Solve for x 2sin(2x)=3cos(2x)

$2 \sin (2 x) = 3 \cos (2 x)$

Step 1

Divide each term in the equation by $\cos (2 x)$ .

$\frac{2 \sin (2 x)}{\cos (2 x)} = \frac{3 \cos (2 x)}{\cos (2 x)}$

Step 2

Separate fractions.

$\frac{2}{1} \cdot \frac{\sin (2 x)}{\cos (2 x)} = \frac{3 \cos (2 x)}{\cos (2 x)}$

Step 3

Convert from $\frac{\sin (2 x)}{\cos (2 x)}$ to $\tan (2 x)$ .

$\frac{2}{1} \cdot \tan (2 x) = \frac{3 \cos (2 x)}{\cos (2 x)}$

Step 4

Divide $2$ by $1$ .

$2 \tan (2 x) = \frac{3 \cos (2 x)}{\cos (2 x)}$

Step 5

Cancel the common factor of $\cos (2 x)$ .

Cancel the common factor.

$2 \tan (2 x) = \frac{3 \cos (2 x)}{\cos (2 x)}$

Divide $3$ by $1$ .

$2 \tan (2 x) = 3$

Step 6

Divide each term in $2 \tan (2 x) = 3$ by $2$ and simplify.

Divide each term in $2 \tan (2 x) = 3$ by $2$ .

$\frac{2 \tan (2 x)}{2} = \frac{3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 \tan (2 x)}{2} = \frac{3}{2}$

Divide $\tan (2 x)$ by $1$ .

$\tan (2 x) = \frac{3}{2}$

Step 7

Take the inverse tangent of both sides of the equation to extract $x$ from inside the tangent.

$2 x = \arctan (\frac{3}{2})$

Step 8

Simplify the right side.

Evaluate $\arctan (\frac{3}{2})$ .

$2 x = 0.98279372$

Step 9

Divide each term in $2 x = 0.98279372$ by $2$ and simplify.

Divide each term in $2 x = 0.98279372$ by $2$ .

$\frac{2 x}{2} = \frac{0.98279372}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{0.98279372}{2}$

Divide $x$ by $1$ .

$x = \frac{0.98279372}{2}$

Simplify the right side.

Divide $0.98279372$ by $2$ .

$x = 0.49139686$

Step 10

The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$2 x = (3.14159265) + 0.98279372$

Step 11

Solve for $x$ .

Add $3.14159265$ and $0.98279372$ .

$2 x = 4.12438637$

Divide each term in $2 x = 4.12438637$ by $2$ and simplify.

Divide each term in $2 x = 4.12438637$ by $2$ .

$\frac{2 x}{2} = \frac{4.12438637}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{4.12438637}{2}$

Divide $x$ by $1$ .

$x = \frac{4.12438637}{2}$

Simplify the right side.

Divide $4.12438637$ by $2$ .

$x = 2.06219318$

Step 12

Find the period of $\tan (2 x)$ .

The period of the function can be calculated using $\frac{π}{| b |}$ .

$\frac{π}{| b |}$

Replace $b$ with $2$ in the formula for period.

$\frac{π}{| 2 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{π}{2}$

Step 13

The period of the $\tan (2 x)$ function is $\frac{π}{2}$ so values will repeat every $\frac{π}{2}$ radians in both directions.

$x = 0.49139686 + \frac{π n}{2}, 2.06219318 + \frac{π n}{2}$ , for any integer $n$

Step 14

Consolidate $0.49139686 + \frac{π n}{2}$ and $2.06219318 + \frac{π n}{2}$ to $0.49139686 + \frac{π n}{2}$ .

$x = 0.49139686 + \frac{π n}{2}$ , for any integer $n$

Solve for x 2sin(2x)=3cos(2x)

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